In recitation 10, slide 14, it's written that "We measure the time of the verifier only in terms of x, not c"
But in lecture 12, slide 55, we ask the question: "Do we always have a small enough certificate?".
Why does it matter?

Well, that was after we described the verifier for the language (IP), note that in particular, the verifier must read all of c,
hence, if c is not polynomial in |x|, then the verifier will certainly not be polynomial in |x|.